COMBINED ESPRIT-ROOTMUSIC FOR DOA-DOD ES-TIMATION IN POLARIMETRIC BISTATIC MIMO RADAR

M. L. Bencheikh and Y. Wang

IREENA Laboratory — SEC Team Polytech’Nantes, Nantes, France

Abstract—In this paper, we investigate the exploitation of the polarimetric diversity signal properties in a bistatic polarimetric MIMO radar to improve the performance of joint estimation of direction of arrival (DOA) and direction of departure (DOD) of targets using combined ESPRIT-RootMUSIC technique. Numerical simulations are carried out to illustrate the performance of the proposed approach.

1. INTRODUCTION

The Multi-Input Multi-Output (MIMO) radar uses multiple antennas to transmit simultaneously several linearly orthogonal waveforms. It also uses multiple antennas to receive the reflected signals [1].

A bistatic MIMO radar scheme has been considered in [2], where a two-dimensional (2-D) spatial spectrum estimation technique based on the Capon method is developed. The proposed method needs an exhaustive search through all the 2-D space to find the DOA and DOD of the targets.

Exploiting the same MIMO radar configuration, Jin et al. [3] have proposed a joint DOA and DOD estimation approach based on the ESPRIT technique. A closed form of the target DOA and DOD is obtained and automatically paired. However, the number of targets which can be localized by this method is smaller than the number of receivers.

With the same architecture of the bistatic MIMO radar, another ESPRIT-based method is proposed in [5] by exploiting the invariance

*Received 3 February 2011, Accepted 23 March 2011, Scheduled 29 March 2011*

property of both the transmitting array and receiving array to estimate the DOA and DOD of targets. The pairing is not obtained automatically by this algorithm, but an additional pairing operation is given.

In [7], we have proposed an approach based on the combination of both the ESPRIT and Root-MUSIC methods to jointly estimate DOA-DOD of targets for a bistatic MIMO radar. This approach allows an automatic pairing between the target’s DOA and DOD.

Polarization diversity have been proven to be useful in array signal processing and various types of radar systems [8, 9, 11, 12].

Recently, by exploiting the invariance property of both the transmitting array and the receiving array, a method based on the ESPRIT technique is proposed in [10], to jointly estimate the parameter DOA-DOD-polarization of target for a polarimetric bistatic MIMO radar, but a pairing operation is needed.

In this work, we propose an extension of the technique developed in [7] by exploiting the polarization diversity in reception antennas. Besides the polarisation diversity, this approach allows an automatic pairing between the target’s DOA and DOD. Simulation results are provided to illustrate the performance of the proposed approach.

2. SIGNAL MODEL

The considered system is a bistatic polarimetric MIMO radar system
with*M* closely spaced transmit antennas and*N* closely spaced receive
antennas. Both of them are uniform linear arrays (ULA) and all the
elements are omnidirectional. ∆tand ∆rare the inter-element spacing
at the transmitter and the receiver, respectively. It is assumed that
the Doppler frequencies have almost no effect on the orthogonality
of the signals. Therefore, the variation of the phase within pulses
caused by Doppler frequency can be ignored. The targets range is
assumed to be much larger than the aperture of transmit array and
receive array. Each element of the transmitter transmits orthogonal
waveforms r*m*, *m* = 1*. . . M. These signals are reflected by a target*

assumed at position (θr*, θ*t) with *θ*t and *θ*r denoting DOD and DOA,
respectively. For *P* targets located at the same range bin, the output
of the entire matched filters at the receiver can be written as [1–3]

z(t) =Cs(t) +n(t) (1)

with the*pth column of the mixing matrix*Cgiven by

c

³

*θ*(* _{r}p*)

*, θ*(

_{t}*p*)

´

=b

³

*θ*(* _{t}p*)

´

*⊗*a

³

*θ*(* _{r}p*)

´¯_{¯}
¯

where

s(t) =

*s*1(t)
..
.
*sP*(t)

_{=}

*α*1*ej*2*πfd*1*t*
..
.
*αPej*2*πfdPt*

(3)

a(θ*r*) =
h

1 *ej*2*π4r*sin(*λθr*) *. . . ej*2*π*(*N−*1)

*4r*sin(*θr*)
*λ*

i_{T}

*∈CN×*1 _{is the }

re-ceiver steering vector; b(θ*t*) =
h

1*ej*2*π4t*sin(*λθt*) *. . . ej*2*π*(*M−*1)
*4 _{t}*sin(

*)*

_{θt}*λ*

i_{T}

*∈CM×*1 _{is the transmitter steering vector;} _{⊗}_{is the Kronecker }
opera-tor; *α* is the reflection coefficient depending on the target radar cross
section; *ej*2*πfdt* _{is due to the target Doppler shift;} _{n}_{(t) is the noise}
vector, whose elements are assumed to be independent, zero mean and
complex Gaussian distributed.

2.1. Polarimetric Model

In order to exploit the polarization diversity in reception, we assume
that the receiver is composed of *N* dual polarization antennas. For
a transverse electromagnetic (TEM) wave incident on the array, the
simplified horizontal and vertical polarization components of the target
reflected signal can be expressed as [8, 9, 12]

z*po*(t) =
·

C(*v*) 0
0 C(*h*)

¸ ·

s(*v*)_{(t)}

s(*h*)_{(t)}

¸

+

·

n(*v*)_{(t)}

n(*h*)_{(t)}

¸

(4)

where *v* and *h* denote the vertical and horizontal polarizations,
respectively, and s(*v*) _{= [s}

1(t) cos(γ1), . . . , s*P*(t) cos(γ*P*)]*T*, s(*h*) =
£

*s*1(t) sin(γ1)e*jξ*1*, . . . , sP*(t) sin(γ*P*)e*jξP*
¤_{T}

are the targets signal vectors
for polarizations *v*and *h, respectively, the noise terms are assumed to*
be uncorrelated and complex Gaussian distributed with polarization
angle *γ* *∈*[0, π/2] and polarization phase difference*ξ* *∈*[*−π, π].*

Assuming that the mixing matrix is polarization-independent, i.e.,

C(*v*)_{(θ}

*r, θt*) = C(*h*)(θ*r, θt*) = C(θ*r, θt*), the extended signal model can

be written as

z*po*(t) =Fs(t) +
·

n(*v*)_{(t)}

n(*h*)_{(t)}

¸

(5)

where F =

·

C 0 0 C

¸ ·

G(*v*)

G(*h*)

¸

, and G(*v*) and G(*h*) are diagonal

matrices with diagonal vectorsg(*v*) _{and} _{g}(*h*) _{given by}

g(*v*) _{=} _{[ cos(γ}

1), . . . ,cos(γ*P*) ]

g(*h*) _{=} £ _{sin(γ}

1)e*jξ*1*, . . . ,*sin(γ*P*)e*jξP*

Then, the covariance matrix of the extended observation vector

z* _{po}* can be expressed as

R*p* =FΓsF*H* +*σ*2I2*M N* (7)

whereΓ_{s} is the covariance matrix ofs(t) and *σ*2 _{is the noise power.}
The directional vector for a single targetfis given by

f(θ*r, θt*) =
·

b(θ*t*)*⊗*a(θ*r*) cos(γ)

b(θ*t*)*⊗*a(θ*r*) sin(γ)e*jξ*
¸

(8)

3. POLARIMETRIC COMBINED ESPRIT-ROOTMUSIC

The MUSIC technique is based on the orthogonality between the noise subspace and the signal subspace. These subspaces can be obtained by the eigenvalues decomposition of the covariance matrix

R*p* *∈ C*2*M N×*2*M N*

R*p* =U*sp*Γ*sp*U*spH* +*σ*2U*np*U*Hnp* (9)

The noise subspace U*np* is orthogonal to the actual target

directional vectors, i.e.,

f(θ*r, θt*)*H*U*np*U*npH*f(θ*r, θt*) = 0, *θr*=*θ*

(*p*)

*r*

*θt*=*θ*(*tp*)
¯
¯
¯
¯
¯

*p*=1*,...,P*

(10)

wheref can be written as

f(θ* _{r}, θ_{t}*) = B(θ

*)*

_{t}·

a(θ*r*) cos(γ)

a(θ*r*) sin(γ)e*jξ*
¸

B(θ*t*) =
·

[b(θ*t*)*⊗*I*N*] 0

0 [b(θ*t*)*⊗*I*N*]
¸

This expression allows to estimate separately the directions.
Therefore, the proposed approach can be subdivided in two steps. In
the first step, we determine only the DOD by the ESPRIT method and
in the second step, for each estimated DOD, the estimated direction
matrix B(ˆ*θt*) is substituted in (10) and the RootMUSIC technique is

applied to estimate the DOA.

According to the principle of ESPRIT technique [5], to exploit
the rotational invariance, in the transmitter array, we define the
submatrices C*t*1 *∈* *C*(*M N−N*)*×P* and C*t*2 *∈* *C*(*M N−N*)*×P*, related to
the DOD, as

C*t*1=

h

b1

³

*θ*(1)_{t}

´

*⊗*a

³

*θ*(1)_{r}

´

*, . . . ,*b1

³

*θ*(* _{t}P*)

´

*⊗*a

³

*θ*(* _{r}P*)

´i

C_{t}_{2}=

h

b_{2}

³

*θ*(1)_{t}

´

*⊗*a

³

*θ*(1)_{r}

´

*, . . . ,*b_{2}

³

*θ*(* _{t}P*)

´

*⊗*a

³

*θ*(* _{r}P*)

withb1(θ(*tp*)) andb2(θ(*tp*)) are the first and the last*M−*1 elements of

b(θ* _{t}*(

*p*)), respectively.

Then, it is easy to show

F*t*2 =F*t*1Φ*t* (12)

whereF_{t}_{1(2)}=

·

C_{t}_{1(2)} 0
0 C_{t}_{1(2)}

¸ ·

G(*v*)
G(*h*)

¸

andΦ*t*is a diagonal matrix

with its main diagonal elements given by *ρ*(* _{t}p*) =

*ej*2

*π4t*sin(

*θ*(

*p*)

*t*)

*λ* for

*p*= 1, . . . , P.

The vectors of U*sp* span the signal subspace, then there exists a

non singular transformation matrix Tsatisfying

U*sp* =FT (13)

Let U*spt*1 and U*spt*2 be the submatrices formed from U*sp* in the
same way as that F*t*1 and F*t*2 are formed from F, we can write

U*spt*1 =F*t*1T*,* U*spt*2 =F*t*2T (14)
Then, the diagonal elements ofΦ*t*are the eigenvalues*ωp*=1*,...,P* of

the following matrix Ω* _{t}* =T

*−*1

_{Φ}

*t*T which satisfies U*spt*2 = U*spt*1Ω*t*,
therefore, the DOD can be estimated by

ˆ

*θ*(* _{t}p*)= arcsin

µ

*λ*

2π∆*t*arg (ω*p*)
¶

(15)

Therefore, for each estimated angle ˆ*θ*(* _{t}p*), if

*θr*is the corresponding

direction of an actual target, then the directional vector is orthogonal to the noise subspace which allows to rewrite the expression (10) as

˙g*H*A*T*(1/z*r*)
·

Π_{n11} Π_{n12}
Πn21 Πn22

¸

A(z*r*)˙g= 0 (16)

whereA(z*r*) =
·

a(z*r*) 0

0 a(z*r*)
¸

,*zr*=*ej*2*π*

*4r*sin(*θr*)
*λ* , ˙g=

·

cos(γ)
sin(γ)e*jξ*

¸

,

and

·

Πn11 Πn12

Πn21 Πn22

¸

=B(ˆ*θ _{t}*)

*H*

_{U}

*np*U*Hnp*B(ˆ*θt*).

The set (16) can be solved by finding the roots of the following polynomial

*det*

µ

A*T*(1/z*r*)
·

Πn11 Πn12

Πn21 Πn22

¸

A(z*r*)
¶

= 0 (17)

which can be expressed by
4_{X}*N−*3

*k*=1

The parameters*q _{k}* are calculated as

*qk*=

min[2_{X}*N−*1*,k*]

*i*=max[1*,k−*2*N*+2]

Q(i, k*−i*+ 1) (19)

whereQ=v11v*T*22*−*v12v*T*21, and the elements ofv*ij* are given by

v* _{ij}*(k) =

min[_{X}*N,N−k*]

*l*=max[1*,*1*−k*]

[Π* _{nij}*]

_{l,k}_{+}

_{l}¯ ¯ ¯ ¯ ¯ ¯

*k*=*−*(*N−*1)*,...,*(*N−*1)

(20)

Therefore, the estimation of the angle *θr* which minimizes the

projection of the directional vector on the noise subspace, is equivalent
to finding the roots of the polynomial (18) for each given ˆ*θt*.

For each estimated DOD, the roots ˆ*zr* inside and closest to

the unitary circle of the polynomial (18) allow to estimate the corresponding DOA angle.

ˆ

*θ*(* _{r}p*) = arcsin

µ

*λ*

2π∆*r* arg (ˆ*zr*)
¶

(21)

We note that, for this proposed method the pairing is automatically obtained between the DOD and DOA angles.

This approach exploiting the polarimetric diversity in bistatic
polarimetric MIMO radar is able to handle more targets than the
classical techniques. The maximum number of targets can be localized
by this approach is 2N(M*−*1).

4. SIMULATION RESULTS

In this section, we investigate the performance of the proposed
approach. For all the simulations, the parameters of targets
((α, f_{d}, γ, ξ) are randomly generated for each target.

Firstly, we consider a bistatic polarimetric MIMO radar composed
of *M* = 3 transmitter elements and *N* = 2 elements in receiver array
spaced by a half wavelength. The data length is*L*= 500 samples and
the number of trials is *K* = 200. For *SN R* = 20 dB, Figure 1 shows
that the proposed approach is able to localize up to*P* = 8 targets.

Secondly, we set *M* = 3, *N* = 4, *L* = 256 and *K* = 200

for two targets (P = 2) located at the angles (θ*r, θt*) = (35*◦,*60*◦*)

(θ*r, θt*) = (80*◦,*15*◦*), Figures 2 and 3 show the Root Mean Square

0 10 20 30 40 50 60 70 80 90 0

10 20 30 40 50 60 70 80 90

DOA (°)

DOD (°

)

Figure 1. Joint DOD-DOA estimation of *P* = 8 targets by the
proposed approach.

0 5 10 15 20 25 30 35 10-3

10-2 10-1 100 101 102

SNR (dB)

RMSE DOA (

°)

ESPRIT RootMUSIC Target 1 ESPRIT RootMUSIC Target 2 Polarimetric ESPRIT RootMUSIC Target 1 Polarimetric ESPRIT RootMUSIC Target 2 CRB Target 1

CRB Target 2

Figure 2. RMSE for DOA estimation versus SNR.

0 5 10 15 20 25 30 35 10-3

10-2 10-1 100 101

SNR (dB)

RMSE DOD (°

)

ESPRIT RootMUSIC Target 2 ESPRIT RootMUSIC Target 1 Polarimetric ESPRIT RootMUSIC Target 2 Polarimetric ESPRIT RootMUSIC Target 1 CRB Target 1

CRB Target 2

Figure 3. RMSE for DOD estimation versus SNR.

Therefore, the exploitation of the polarization diversity in
reception improves considerably the estimation performance because
the observations space is extended to 2M N *×* 2M N from *M N* *×*

*M N* which increases also the number of localizable targets and the
resolution power.

5. CONCLUSION

We have exploited the additional degrees-of-freedom of the polarization diversity to improve the performances of the combined ESPRIT-RootMUSIC proposed in [7] for joint DOA-DOD estimation in a bistatic MIMO radar. The exploitation of the polarimetric diversity allows to handle more targets than the classical algorithms and improve the localization performances.

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